Congestion Games

Kevin Leyton-Brown

Congestion Games Kevin Leyton-Brown, Slide 1

What you’ll learn in this talk

What is a congestion game?

. . . and, that there are other ways of representingsimultaneous-move games

What sorts of interactions do they model?

What good theoretical properties do they have?

What are potential games, and how are they related tocongestion games?

Congestion Games Kevin Leyton-Brown, Slide 2

Definition

Each player chooses some subset from a set of resources, and the cost of eachresource depends on the number of other agents who select it.

Definition (Congestion game)

A congestion game is a tuple (N,R,A, c), where

N is a set of n agents;

R is a set of r resources;

A = A1 × · · · ×An, where Ai ⊆ 2R \ {∅} is the set of actions for agent i;

c = (c1, . . . , cr), where ck : N 7→ R is a cost function for resource k ∈ R.

Utility functions:

Define # : R×A 7→ N as a function that counts the number of playerswho took any action that involves resource r under action profile a.

For each resource k, define a cost function ck : N 7→ R.

Given an action profile a = (ai, a−i),

ui(a) = −∑

r∈R|r∈ai

cr(#(r, a)).

Congestion Games Kevin Leyton-Brown, Slide 3

Definition

Each player chooses some subset from a set of resources, and the cost of eachresource depends on the number of other agents who select it.

Definition (Congestion game)

A congestion game is a tuple (N,R,A, c), where

N is a set of n agents;

R is a set of r resources;

A = A1 × · · · ×An, where Ai ⊆ 2R \ {∅} is the set of actions for agent i;

c = (c1, . . . , cr), where ck : N 7→ R is a cost function for resource k ∈ R.

Utility functions:

Define # : R×A 7→ N as a function that counts the number of playerswho took any action that involves resource r under action profile a.

For each resource k, define a cost function ck : N 7→ R.

Given an action profile a = (ai, a−i),

ui(a) = −∑

r∈R|r∈ai

cr(#(r, a)).

Congestion Games Kevin Leyton-Brown, Slide 3

Definition

Each player chooses some subset from a set of resources, and the cost of eachresource depends on the number of other agents who select it.

Definition (Congestion game)

A congestion game is a tuple (N,R,A, c), where

N is a set of n agents;

R is a set of r resources;

A = A1 × · · · ×An, where Ai ⊆ 2R \ {∅} is the set of actions for agent i;

c = (c1, . . . , cr), where ck : N 7→ R is a cost function for resource k ∈ R.

Utility functions:

Define # : R×A 7→ N as a function that counts the number of playerswho took any action that involves resource r under action profile a.

For each resource k, define a cost function ck : N 7→ R.

Given an action profile a = (ai, a−i),

ui(a) = −∑

r∈R|r∈ai

cr(#(r, a)).

Congestion Games Kevin Leyton-Brown, Slide 3

Motivating Example: Selfish Routing

Agents trying to choose uncongested paths in a graph.

Each edge connecting two nodes is a resourceActions are paths in the graph that connect a given user’ssource and target nodes

Stream a video in a computer networkTravel along a road network

The cost function for each resource expresses the latency oneach link as a function of its congestion

an increasing (possibly nonlinear) function

Congestion Games Kevin Leyton-Brown, Slide 4

Motivating Problem: Santa Fe (“El Farol”) Bar Problem

Each of a set of people independently selects whether or notto go to the bar

Utility for attending:number of people attending, if less than or equal to 6;6 minus the number of people attending, if greater than 6

Utility for not attending: 0

Note: nonmonotonic cost functions

Congestion Games Kevin Leyton-Brown, Slide 5

Motivating Problem: Santa Fe (“El Farol”) Bar Problem

Each of a set of people independently selects whether or notto go to the barUtility for attending:

number of people attending, if less than or equal to 6;6 minus the number of people attending, if greater than 6

Utility for not attending: 0Note: nonmonotonic cost functions

Play the game (raising your hands)Congestion Games Kevin Leyton-Brown, Slide 5

Motivating Problem: Santa Fe (“El Farol”) Bar Problem

Each of a set of people independently selects whether or notto go to the barUtility for attending:

number of people attending, if less than or equal to 6;6 minus the number of people attending, if greater than 6

Utility for not attending: 0

Note: nonmonotonic cost functions

Play again. Change your action if you like.Congestion Games Kevin Leyton-Brown, Slide 5

Represent the Santa Fe Bar Problem as a Congestion Game

Definition (Congestion game)

A congestion game is a tuple (N,R,A, c), where

N is a set of n agents;

R is a set of r resources;

A = A1 × · · · ×An, where Ai ⊆ 2R \ {∅} is the set of actions for agent i;

c = (c1, . . . , cr), where ck : N 7→ R is a cost function for resource k ∈ R.

Utility functions:

Define # : R×A 7→ N as a function that counts the number of playerswho took any action that involves resource r under action profile a.

For each resource k, define a cost function ck : N 7→ R.

Given an action profile a = (ai, a−i),

ui(a) = −∑

r∈R|r∈ai

cr(#(r, a)).

Congestion Games Kevin Leyton-Brown, Slide 6

Why care about congestion games?

Theorem

Every congestion game has a (“pure-strategy”) Nash equilibrium.

Theorem

A simple procedure (MyopicBestResponse) is guaranteed tofind a pure-strategy Nash equilibrium of a congestion game.

Congestion Games Kevin Leyton-Brown, Slide 7

Myopic Best Response

Start with an arbitrary action profile a

While there exists an agent i for whom ai is not a bestresponse to a−i

a′i ← some best response by i to a−ia← (a′i, a−i)

Return a

By the definition of equilibrium, MyopicBestResponse returnsa pure-strategy Nash equilibrium if it terminates.

Congestion Games Kevin Leyton-Brown, Slide 8

Analyzing MBR

In general games MyopicBestResponse can get caught in acycle, even when a pure-strategy Nash equilibrium exists.

L C R

U −1, 1 1,−1 −2,−2

M 1,−1 −1, 1 −2,−2

D −2,−2 −2,−2 2, 2

Can you find a cycle?

Congestion Games Kevin Leyton-Brown, Slide 9

Analyzing MBR

In general games MyopicBestResponse can get caught in acycle, even when a pure-strategy Nash equilibrium exists.

L C R

U −1, 1 1,−1 −2,−2

M 1,−1 −1, 1 −2,−2

D −2,−2 −2,−2 2, 2

This game has one pure-strategy Nash equilibrium, (D,R).However, if we run MyopicBestResponse with a = (L,U) theprocedure will cycle forever.

Congestion Games Kevin Leyton-Brown, Slide 9

Potential Games

Definition (Potential game)

A game G = (N,A, u) is a potential game if there exists afunction P : A 7→ R such that, for all i ∈ N , all a−i ∈ A−i andai, a

′i ∈ Ai, ui(ai, a−i)− ui(a

′i, a−i) = P (ai, a−i)− P (a′i, a−i).

Theorem

Every potential game has a pure-strategy Nash equilibrium.

Proof.

Let a∗ = argmaxa∈A P (a). Clearly for any other action profile a′,P (a∗) ≥ P (a′). Thus by the definition of a potential function, forany agent i who can change the action profile from a∗ to a′ bychanging his own action, ui(a

∗) ≥ ui(a′).

Congestion Games Kevin Leyton-Brown, Slide 10

Potential Games

Definition (Potential game)

A game G = (N,A, u) is a potential game if there exists afunction P : A 7→ R such that, for all i ∈ N , all a−i ∈ A−i andai, a

′i ∈ Ai, ui(ai, a−i)− ui(a

′i, a−i) = P (ai, a−i)− P (a′i, a−i).

Theorem

Every potential game has a pure-strategy Nash equilibrium.

Proof.

Let a∗ = argmaxa∈A P (a). Clearly for any other action profile a′,P (a∗) ≥ P (a′). Thus by the definition of a potential function, forany agent i who can change the action profile from a∗ to a′ bychanging his own action, ui(a

∗) ≥ ui(a′).

Congestion Games Kevin Leyton-Brown, Slide 10

Potential Games

Definition (Potential game)

A game G = (N,A, u) is a potential game if there exists afunction P : A 7→ R such that, for all i ∈ N , all a−i ∈ A−i andai, a

′i ∈ Ai, ui(ai, a−i)− ui(a

′i, a−i) = P (ai, a−i)− P (a′i, a−i).

Theorem

Every potential game has a pure-strategy Nash equilibrium.

Proof.

Let a∗ = argmaxa∈A P (a). Clearly for any other action profile a′,P (a∗) ≥ P (a′). Thus by the definition of a potential function, forany agent i who can change the action profile from a∗ to a′ bychanging his own action, ui(a

∗) ≥ ui(a′).

Congestion Games Kevin Leyton-Brown, Slide 10

Congestion Games have Pure-Strategy Equilibria

Theorem

Every congestion game is a potential game.

Every congestion game has the potential function

P (a) =∑r∈R

#(r,a)∑j=1

cr(j)

There’s a proof in the bookMain intuition: utility functions are linear combinations of costfunctions, so most of the terms in this expansion cancel outwhen we take the difference between the potential values fortwo similar action profiles

It also turns out that every potential game is a congestiongame (harder to show)

Congestion Games Kevin Leyton-Brown, Slide 11

Myopic Best Response Converges

Theorem

The MyopicBestResponse procedure is guaranteed to find apure-strategy Nash equilibrium of a congestion game.

Proof.

It is sufficient to show that MyopicBestResponse finds apure-strategy Nash equilibrium of any potential game. With everystep of the while loop, P (a) strictly increases, because byconstruction ui(a

′i, a−i) > ui(ai, a−i), and thus by the definition of

a potential function P (a′i, a−i) > P (ai, a−i). Since there are onlya finite number of action profiles, the algorithm must terminate.

Congestion Games Kevin Leyton-Brown, Slide 12

Myopic Best Response: Analysis

MyopicBestResponse converges for CGs regardless of:

the cost functions (e.g., they do not need to be monotonic)

the action profile with which the algorithm is initialized

which agent best responds (when there’s a choice)

And even if we change best response to “better response”

Complexity considerations:

the problem of finding a pure Nash equilibrium in a congestiongame is PLS-complete

as hard to find as any other object whose existence isguaranteed by a potential function argumentintuitively, as hard as finding a local minimum in a travelingsalesman problem using local search

We thus expect MyopicBestResponse to be inefficient inthe worst case

Congestion Games Kevin Leyton-Brown, Slide 13

Myopic Best Response: Analysis

MyopicBestResponse converges for CGs regardless of:

the cost functions (e.g., they do not need to be monotonic)

the action profile with which the algorithm is initialized

which agent best responds (when there’s a choice)

And even if we change best response to “better response”

Complexity considerations:

the problem of finding a pure Nash equilibrium in a congestiongame is PLS-complete

as hard to find as any other object whose existence isguaranteed by a potential function argumentintuitively, as hard as finding a local minimum in a travelingsalesman problem using local search

We thus expect MyopicBestResponse to be inefficient inthe worst case

Congestion Games Kevin Leyton-Brown, Slide 13

Conclusions

Congestion games are a compact and intuitive way ofrepresenting interactions in which agents care about thenumber of others who choose a given resource, and theirutility decomposes additively across these resources

Potential games are a less-intuitive but analytically usefulcharacterization equivalent to congestion games

potential function: a single function that captures any player’sutility change from deviating

These games always have pure-strategy Nash equilibria

MyopicBestResponse always converges to a pure-strategyNash equilibrium in congestion/potential games.

They’re very widely studied in the literature, particularly in CSrealistic modelpure-strategy equilibria are actually fairly rarenice computational story

Congestion Games Kevin Leyton-Brown, Slide 14

References

1 A class of games possessing pure-strategy Nash equilibria, RWRosenthal, International Journal of Game Theory, 1973.

2 Potential games, D Monderer, LS Shapley, Games andEconomic Behavior, 1996.

3 Bounding the inefficiency of equilibria in nonatomiccongestion games, T Roughgarden, Tardos, Games andEconomic Behavior, 2004.

4 “The El Farol Bar Problem”, Wikipedia,http://en.wikipedia.org/wiki/El_Farol_Bar_problem,accessed 1/15/2014.

5 Multiagent systems: Algorithmic, game-theoretic, and logicalfoundations, Y Shoham, K Leyton-Brown, CambridgeUniversity Press, 2009, §6.4.

Congestion Games Kevin Leyton-Brown, Slide 15